Circle Endpoint Calculator

The Circle Endpoint Calculator is a versatile tool designed to calculate specific points on a circle’s boundary based on inputs like the center, radius, or diameter endpoints. This is especially helpful in geometry, graphic design, and computational modeling. With this calculator, you can easily find endpoints without manual calculations, ensuring speed and accuracy in your projects.

Formula of Circle Endpoint Calculator

The method for calculating endpoints depends on the available inputs.

Case 1: Using Center and Radius

If the circle’s center is given as (h, k) and the radius is r, the endpoints can be determined by moving r units in specific directions.

Formulas for endpoints:

1. (h + r, k) – Endpoint to the right
2. (h - r, k) – Endpoint to the left
3. (h, k + r) – Endpoint upward
4. (h, k - r) – Endpoint downward

Case 2: Using Diameter Endpoints

If you know the endpoints of a circle’s diameter, you can derive the center and radius to find other circle properties.

1. Midpoint Formula
   Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
   • This gives the center of the circle.
2. Distance Formula
   Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
   • The radius is half the distance between the diameter endpoints.

Once the center and radius are known, use the formulas from Case 1 to calculate specific endpoints.

A Table for Common Conversions and Terms

Here’s a reference table for common circle configurations and their calculated endpoints:

Center (h, k)	Radius (r)	Endpoints
(0, 0)	5	(5, 0), (-5, 0), (0, 5), (0, -5)
(3, 4)	6	(9, 4), (-3, 4), (3, 10), (3, -2)
(-2, -2)	3	(1, -2), (-5, -2), (-2, 1), (-2, -5)

This table demonstrates how endpoints shift with different center and radius values.

Example of Circle Endpoint Calculator

Let’s calculate endpoints using an example:

Problem

A circle has its center at (2, 3) and a radius of 4 units. Find the endpoints.

Solution

Using the endpoint formulas:

1. Right endpoint: (h + r, k) = (2 + 4, 3) = (6, 3)
2. Left endpoint: (h - r, k) = (2 - 4, 3) = (-2, 3)
3. Upper endpoint: (h, k + r) = (2, 3 + 4) = (2, 7)
4. Lower endpoint: (h, k - r) = (2, 3 - 4) = (2, -1)

Thus, the endpoints are (6, 3), (-2, 3), (2, 7), and (2, -1).

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